Question

Consider the surface whose equation is z = -2(y+1)^2 + (x+1)^2 - 1

i) determine a surface of the form (*) whose graph you can translate, rotate, or reflect to obtain the graph of the given surface. Provide the steps one would follow to manipulate the graph of your surface to get the graph of the given surface.

Form could be Ellipsoid, Cone, Elliptic Paraboloid, Hyperboloid of One sheet, Hyperbolic paraboloid, and Hyperboloid of two sheets.

Answer 1

Consider the surface whose equation is z = -2y² + x² - 1 which is an Hyperbolic paraboloid. This surface when translated through one unit in y- direction and one unit in x- direction gives the given surface of equation z = -2(y+1)² + (x+1)² - 1.

First translate through one unit in y- direction thus we obtain

6 4 2 0 6 2 囲 a(x, y) =-2y2+8-1 b(x, y) =-2(y + 1)2 + x2-1

Now translate through one unit in x-direction and we obtain

o0 -5 O b(x,y)=-2(y + 1)2 + x2-1

Thus our surface is translated to the given surface

15 -5 -10 10 a(x, y) =-2(y + 1)2 + (x + 1)2-1 b(x, y) =-2y2HX2-1